scholarly journals The Tilted Flashing Brownian Ratchet

2019 ◽  
Vol 18 (01) ◽  
pp. 1950005 ◽  
Author(s):  
S. N. Ethier ◽  
Jiyeon Lee
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Planck Equation ◽  
Directed Motion ◽  
Brownian Ratchet ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Motion Effect ◽  
Viable Method ◽  
Direction Opposite

The flashing Brownian ratchet is a stochastic process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, the latter being a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. The result is directed motion. In the presence of a static homogeneous force that acts in the direction opposite to that of the directed motion, there is a reduction (or even a reversal) of the directed motion effect. Such a process may be called a tilted flashing Brownian ratchet. We show how one can study this process numerically, using a random walk approximation or, equivalently, using numerical solution of the Fokker–Planck equation. Stochastic simulation is another viable method.

scholarly journals The flashing Brownian ratchet and Parrondo’s paradox

2018 ◽  
Vol 5 (1) ◽  
pp. 171685 ◽  
Author(s):  
S. N. Ethier ◽  
Jiyeon Lee
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Diffusion Process ◽  
Directed Motion ◽  
Time And Space ◽  
Brownian Ratchet ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Games Of Chance ◽  
Parrondo’S Paradox

A Brownian ratchet is a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. A flashing Brownian ratchet is a process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, producing directed motion. These processes have been of interest to physicists and biologists for nearly 25 years. The flashing Brownian ratchet is the process that motivated Parrondo’s paradox, in which two fair games of chance, when alternated, produce a winning game. Parrondo’s games are relatively simple, being discrete in time and space. The flashing Brownian ratchet is rather more complicated. We show how one can study the latter process numerically using a random walk approximation.

The Level-Crossing Problem: First-Passage, Escape and Extremes

2014 ◽  
Vol 13 (04) ◽  
pp. 1430001 ◽  
Author(s):  
Jaume Masoliver
Keyword(s):  
Brownian Motion ◽  
Diffusion Processes ◽  
Extreme Values ◽  
Level Crossing ◽  
First Passage ◽  
One Dimensional ◽  
Absolute Value ◽  
Dimensional Diffusion

We review the level-crossing problem which includes the first-passage and escape problems as well as the theory of extreme values (the maximum, the minimum, the maximum absolute value and the range or span). We set the definitions and general results and apply them to one-dimensional diffusion processes with explicit results for the Brownian motion and the Ornstein–Uhlenbeck (OU) process.

On the solution of the Fokker–Planck equation for a Feller process

1990 ◽  
Vol 22 (01) ◽  
pp. 101-110
Author(s):  
L. Sacerdote
Keyword(s):  
Diffusion Processes ◽  
Planck Equation ◽  
Hyperbolic Type ◽  
Diffusion Equations ◽  
Time Varying ◽  
Feller Process ◽  
Parameter Group ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Type Boundary

Use of one-parameter group transformations is made to obtain the transition p.d.f. of a Feller process confined between the origin and a hyperbolic-type boundary. Such a procedure, previously used by Bluman and Cole (cf., for instance, [4]), although useful for dealing with one-dimensional diffusion processes restricted between time-varying boundaries, does not appear to have been sufficiently exploited to obtain solutions to the diffusion equations associated to continuous Markov processes.

On the solution of the Fokker–Planck equation for a Feller process

1990 ◽  
Vol 22 (1) ◽  
pp. 101-110 ◽  
Author(s):  
L. Sacerdote
Keyword(s):  
Diffusion Processes ◽  
Planck Equation ◽  
Hyperbolic Type ◽  
Diffusion Equations ◽  
Time Varying ◽  
Feller Process ◽  
Parameter Group ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Type Boundary

Use of one-parameter group transformations is made to obtain the transition p.d.f. of a Feller process confined between the origin and a hyperbolic-type boundary. Such a procedure, previously used by Bluman and Cole (cf., for instance, [4]), although useful for dealing with one-dimensional diffusion processes restricted between time-varying boundaries, does not appear to have been sufficiently exploited to obtain solutions to the diffusion equations associated to continuous Markov processes.

Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

2007 ◽  
Vol 44 (02) ◽  
pp. 393-408 ◽  
Author(s):  
Allan Sly
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
White Noise ◽  
Gaussian Process ◽  
Discrete Data ◽  
Hölder Exponent ◽  
Scaling Properties ◽  
Multifractional Brownian Motion ◽  
Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

The Fluctuating Lattice Boltzmann

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Brownian Motion ◽  
Fluid Flow ◽  
Lattice Boltzmann ◽  
Thermal Fluctuations ◽  
Directed Motion ◽  
Statistical Fluctuations ◽  
Lattice Boltzmann Models ◽  
Motion Versus ◽  
The Way

Fluid flow at nanoscopic scales is characterized by the dominance of thermal fluctuations (Brownian motion) versus directed motion. Thus, at variance with Lattice Boltzmann models for macroscopic flows, where statistical fluctuations had to be eliminated as a major cause of inefficiency, at the nanoscale they have to be summoned back. This Chapter illustrates the “nemesis of the fluctuations” and describe the way they have been inserted back within the LB formalism. The result is one of the most active sectors of current Lattice Boltzmann research.

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